KAM Theorem of Symplectic Algorithms

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

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A New KAM Theorem for the Hyperbolic Lower Dimensional Tori in Reversible Systems

Acta Applicandae Mathematicae ◽

10.1007/s10440-015-0027-0 ◽

2015 ◽

Vol 143 (1) ◽

pp. 45-61 ◽

Cited By ~ 1

Author(s):

Xiaocai Wang ◽

Junxiang Xu ◽

Dongfeng Zhang

Keyword(s):

Reversible Systems ◽

Kam Theorem ◽

Lower Dimensional Tori ◽

Lower Dimensional

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The Kolmogorov-Arnold-Moser (KAM) Theorem

Fundamental Principles of Classical Mechanics ◽

10.1142/9789814551496_0041 ◽

2014 ◽

pp. 487-520

Keyword(s):

Kam Theorem

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The KAM Theorem

Classical and Quantum Dynamics ◽

10.1007/978-3-030-36786-2_16 ◽

2020 ◽

pp. 197-204

Author(s):

Walter Dittrich ◽

Martin Reuter

Keyword(s):

Kam Theorem

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The KAM theorem and renormalization group

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080450 ◽

2009 ◽

Vol 29 (2) ◽

pp. 419-431 ◽

Cited By ~ 5

Author(s):

E. DE SIMONE ◽

A. KUPIAINEN

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Renormalization Group ◽

Elementary Proof ◽

Renormalization Group Equation ◽

Quadratic Nonlinearity ◽

Picard Iteration ◽

Group Equation ◽

Quantum Field ◽

Kam Theorem

AbstractWe give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a certain PDE with quadratic nonlinearity, the so-called Polchinski renormalization group equation studied in quantum field theory.

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Superconvergent Perturbation Theory, KAM Theorem (Introduction)

Classical and Quantum Dynamics - Advanced Texts in Physics ◽

10.1007/978-3-642-56430-7_14 ◽

2001 ◽

pp. 155-162

Author(s):

W. Dittrich ◽

Martin Reutera

Keyword(s):

Perturbation Theory ◽

Kam Theorem

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Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415500431 ◽

2015 ◽

Vol 25 (03) ◽

pp. 1550043 ◽

Cited By ~ 2

Author(s):

Yixian Gao ◽

Weipeng Zhang ◽

Shuguan Ji

Keyword(s):

Wave Equation ◽

Normal Form ◽

Periodic Solutions ◽

Nonlinear Wave ◽

Nonlinear Wave Equation ◽

Seismic Waves ◽

Birkhoff Normal Form ◽

General Boundary Conditions ◽

Kam Theorem ◽

Infinite Dimensional

This paper is devoted to the study of quasi-periodic solutions of a nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Based on the partial Birkhoff normal form and an infinite-dimensional KAM theorem, we can obtain the existence of quasi-periodic solutions for this model under the general boundary conditions.

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